Download - Resonant superfluidity in a dilute Fermi gas
Ramsey fringes in a Ramsey fringes in a BoseBose--Einstein Einstein condensate between atoms and condensate between atoms and
moleculesmolecules
Servaas KokkelmansENS, Paris
Collaboration:Theory ExperimentMurray Holland Neil ClaussenJosh Milstein Liz DonleyMarilu Chiofalo Carl Wieman
JILA, University of Colorado and NIST
AtomAtom--molecule coherencemolecule coherenceRecent experiment at JILA with 85Rb condensate:
Feshbach resonance causes coherent coupling Atoms molecules Donley et al., Nature 412 295 (2002).
Apply two field-pulses close to resonance
0 20 40 60 80 100154
155
156
157
158
159
160
161
162
163
t (µs)
B (G
auss
)
tevolve
t (µs)
B (G
auss
) tevolve
0 155 300
attractive
B (G)
-∞
a<0
Nmax=80
+∞ repulsivea>0
a (a
0)
What happens to BEC?What happens to BEC?
Expanded BEC, no B-field pulseN0 ~ 17,000
480 µm
In trap focused burst atoms(150 nK) Nburst/N0 = 25% - 40%
Cold < 3 nK BEC remnantNrem/N0 = 65% - 25%After B-field pulse,
See 2 components
Also missing atoms…………..
AtomAtom--molecule coherencemolecule coherenceTwo observed components oscillate!
RemnantBurst
10 15 20 25 30 35 400
4
8
12
16
tevolve (µs)
Num
ber (
x103 )
Looks likeRamsey-Fringes!
Molecular stateMolecular stateOscillations correspond to binding energy Feshbachmolecular state
Molecules play an important role close to resonance!
Coupled channels calculationUsed analysis from Kempen, Kokkelmans, Verhaar, Phys. Rev. Lett. 88, 093201 (2002)
Simple model
2
2
2)(
aBE
µη
=B (Gauss)
What is What is Feshbach Feshbach resonance?resonance?Coupling between open and closed channels:
Separate out bound state and treat explicitly
Resonance: short-range molecular stateRelatively long-lived moleculesScattering becomes strongly energy-dependent
closed channel
open channel
a
B
abg
Ekin
Resonance scattering: no GP equationResonance scattering: no GP equationClose to resonance, pairing field is important
Scattering length a large, na3 > 1Correlations induced by molecular stateEnergy-dependent scatteringInclude explicitly short-range molecular state in Hamiltonian
Describe two-body interaction with few parameters:Sc
atte
ring
leng
th
Detuning v
Width g
abg
Resonance HamiltonianResonance HamiltonianSplit interactions into two parts:
Direct non-resonant interaction (background process)Resonance coupling to intermediate molecular state
with
( )
( )]..)()()()(
)()()()()([
)()()()()()(
121212
12122123
13
3
cHxxxgX
xxxVxxxdxd
xxHxxxHxxdH
aam
aaaa
mmmaaa
++
+
+=
+
++
++
∫∫
ψψψ
ψψψψ
ψψψψ
mxHa 2/)( 22∇−= η 2112 xxx −=2/)( 2112 xxX +=ν+∇−= mxHm 4/)( 22η
and V(x12) and g(x12) contact interactions
Field equationsField equationsHartree-Fock-Bogoliubov approx.: Define mean-fields
Hartree-Fock-Bogoliubov approx. gives rise to coupled field equations:
,aa ψφ = ,mm ψφ = ,aaNG χχ += ,aaAG χχ=
))()(2]())0(([
)())0(|(|4)(2
)(
)]()())0(([Im2)(
))0((2
))0(())0(2|(|
2
222
**2
2
*2
rrGgGV
rGGVrGdtrdGi
rGgrGGVdtrdG
Ggdtdi
gVGGVdtdi
NmAa
ANaAA
AmAAaN
maAam
amAaNaa
δφφ
φµ
φφ
νφφφφ
φφφφφ
++++
++∇
−=
++=
++=
+++=
ηη
η
η
ηatomic condensate
molecular condensate
normal field
anomalous field
Resonance scattering equations insideResonance scattering equations insideSetting density-dependent terms to zero
Get coupled two-body scattering eqns.
Energy-dependent scattering close to resonanceContact interaction gives rise to divergence in k-space
mm
m
Pgdtdi
rgrPrVdtrdPi
νφφ
φδδµ
+=
+
+
∇−=
)0(2
)()()(2
)( 22
η
ηη
See PRA 65, 053617 (2002)
How to resolve this?Renormalize equations
Get the 2Get the 2--body physics rightbody physics rightSteps involved to get to renormalized resonance scattering theory:
Full CC scattering
Feshbach model
Analytic square-well
Renormalized scattering
Feshbach Feshbach theorytheoryShows that only few parameters needed to describe full energy-
dependent scattering:Coupling open en closed channels
Resonant S-matrix
T-matrix
Zero limit:
scattering length:
closed channel
open channel
−−
−= −
)(21)( 242
22
kinm
ika
EikgikgekS bg
νπη
[ ]1)(2)( −= kSkikT ηπ
)0()(424
22
→− kgam m
bg νπ
πη
ηScattering Energy
Re[
T]-m
atrix
ν
Contact scattering Contact scattering -- renormalizationrenormalizationLimiting case: R 0
Cut-off gives renormalization!Define parameters
∑∫
∫
−
−+
−=
i ikin
K
iii
K
V
E
dpkTgcg
dpkTVcVkT
cutoff
cutoff
ν0
0
)(
)()(Solve Lippmann-Schwinger equation with contact potentials and contact coupling:
Uα−=Γ
11
222 ηπα
cutoffmK=
bgamU
24 ηπ=
UΓ=U
11g gΓ=
1111 ggανν +=
Relation between “real” and cut-off parameters:
(for single resonance)
Simulation experimentSimulation experimentSolve resonance theory for experimental conditions :
0 20 40 60 80 1000
0.005
0.01
0.015
0 20 40 60 80 1000.4
0.5
0.6
0.7
0.8
0.9
1
t (µs)
Atomic condensatefraction
Oscillations at binding-energy frequency!
t (µs)
2|| aφ
2|| mφ
Molecular condensatefraction
Binding energyBinding energyOscillation frequency agrees with molecular binding energy:
158 158.5 159 159.5 160 160.5 1610
50
100
150
200
250
300
350
400
450
500
B (Gauss)
E B(k
Hz)
Oscillations
Coupled channels
Simulation experiment (2)Simulation experiment (2)Crucial aspect:
Growth of non-condensate component!Oscillates almost out of phase with atomic condensateNot a usual thermal gas: coherent because of rise pairing field GA
GN(r) is correlation functionCan determine temperature of these atoms:Is consistent with experiment
Ramsey FringesRamsey Fringes
10 15 20 25 30 35 400
2000
4000
6000
8000
10000
12000
14000
16000
18000
t (µs)
Num
ber
Simulate experiment for different tevolve:Correct visibility, mean valueCorrect oscillation frequencySame (small) phase-shift as in experiment
Identify different fractions as:
Remnant atomic condensate
Burst atoms coherent non-condensate
Missing atoms atoms in molecular state
tevolve (µs)
Num
ber
Atomic condensate
Non-condensate
160 µs
Change pulse shapeChange pulse shape
Longer fall time
10 µs
155 G
tevolve (µs)
2 4 6 8 10 12 14 16
Num
ber
0
4000
8000
12000
16000
tevolve (µs)10 15 20 25 30 35 40
Num
ber (
x10-3
)
0
4
8
12
16
short fall time
long fall time
Remnant+burst
Remnant
Burst
More molecules!More molecules!
Phase shift smaller, so much bigger oscillationsin total number of observed atoms.
Precision binding energy measurementPrecision binding energy measurement
oscillation freq.+
B-field (pulsed NMR)
tevolve (µs)10 20 30 40
Rem
nant
Num
ber (
x10-3
)
9
10
11
12
13
14
ν=196.6(11) kHz
Bound statespectroscopy
Improving the Improving the interactomic interactomic potentialspotentials
B (G)156 157 158 159 160 161 162
Freq
uenc
y (k
Hz)
10
100
1000
Ingredients:6 most accurate oscillation frequenciesPosition of zero crossing scattering length (a=0): B’=165.75(1)
Very close to thresholdIn agreement with previous 87Rb-85Rb determinationUncertainty in B0 reduced by factor 10
abg = -450.5 +- 4 a0
B0= 154.95 +- 0.03 G
25 50 75 100-500-250
0250
12840
12880
12920
12960
ener
gy (c
m-1)
internuclear distance (a0)
Og- (5P3/2)
ground state (5S1/2)
v = 0 - 10
νmolecule
- Scott Papp, Sarah Thompson, Carl Wieman
How to detect the moleculesHow to detect the molecules
Short laser pulse
Minimize photoassociationof BEC (B-field, laser freq).
Look for bound-bound transitions.
SternStern--GerlachGerlach separatorseparator
µdimer strong function of B near resonance
Choose Bevolve where dimers are untrapped
After pulse #1, wait for dimers to fall Apply 2nd pulse, look for position shift
of atoms
E
µ > 0 µ < 0
Other possibility: Large detuning (for optical trap), blow away atomsMolecules remain
B
ConclusionsConclusionsExplain observed coherent oscillations atoms-molecules in 85Rbcondensate
Pairing field plays crucial role, gives rise to coherent non-condensate atoms
Non-condensate larger than molecular condensate
Agreement also for different densities
Based on formulation of resonance pairing model by separating out highest bound states
Resonance scattering built-in in many-body theory: coupled channels with contact potentials
High precision bound state spectroscopy improves potentials
Previously used for description of resonance superfluidity
PRL 89, 180401 (2002), PRL 87, 120406 (2002), PRA 65, 053617 (2002)
Double square wellDouble square wellSimple model to describe Feshbach resonance
Coupled square well
Range R 0: Contact potentials
-V1-V2
Potential range R
ε
Ekin0
Detuning
Simple wave-Functions:Molecular and “free”
Coupling:Boundarycondition
uP(r), uQ(r)
u1(r), u2(r)
−
=
)()(
cossinsincos
)()(
2
1
RuRu
RuRu
Q
P
θθθθ
Can do better:Can do better: 66Li Li FeshbachFeshbach resonanceresonance
B (Gauss)
a (U
nits
of a
0)
Two lowest hyperfine states (1/2,1/2)+(1/2,-1/2)
Double resonance!
Double-resonance S-matrix:
With ,
And coupling strengths g1 and g2
Realbackground
!31 0aabg =
∆∆−∆+∆
∆+∆−= −
211222
21
1222
212
)()(21)(
ggikggikekS bgika
mkinE24
11 )( ηπν −=∆ mkinE24
22 )( ηπν −=∆
Double res. needed for binding energyDouble res. needed for binding energyCompare different models for calculation of binding energy (85Rb)
0.0155 0.0156 0.0157 0.0158 0.0159 0.0161
-600
-500
-400
-300
-200
-100 Single res.
2
2
maE η=
Simple contactmodel
Coupled channels
Double res.Eff. range
Bin
ding
ene
rgy
B (T)
More interesting structure arisesMore interesting structure arises
0.0155 0.016 0.0165 0.017 0.0175 0.018
-30000
-20000
-10000
10000
20000
30000
Double resonance model shows also quasi-bound state:
Also virtual states arise:Work in progress!
Bin
ding
ene
rgy
B (T)